Average Error: 11.8 → 0.4
Time: 7.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\sqrt[3]{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)\]
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\frac{\frac{2}{r}}{r} - \left(\sqrt[3]{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)
double code(double v, double w, double r) {
	return ((double) (((double) (((double) (3.0 + ((double) (2.0 / ((double) (r * r)))))) - ((double) (((double) (((double) (0.125 * ((double) (3.0 - ((double) (2.0 * v)))))) * ((double) (((double) (((double) (w * w)) * r)) * r)))) / ((double) (1.0 - v)))))) - 4.5));
}
↓
double code(double v, double w, double r) {
	return ((double) (((double) (((double) (2.0 / r)) / r)) - ((double) (((double) (((double) cbrt(((double) pow(((double) (((double) (0.125 * ((double) (3.0 - ((double) (2.0 * v)))))) / ((double) (1.0 - v)))), 3.0)))) * ((double) (((double) (((double) fabs(w)) * r)) * ((double) (((double) fabs(w)) * r)))))) + ((double) (4.5 - 3.0))))));
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 11.8

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]

  2. Simplified7.7

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, 4.5\right) - 3\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt35.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot \color{blue}{\left(\sqrt{r} \cdot \sqrt{r}\right)}, 4.5\right) - 3\right)\]

  5. Applied add-sqr-sqrt35.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\sqrt{r} \cdot \sqrt{r}\right)}\right) \cdot \left(\sqrt{r} \cdot \sqrt{r}\right), 4.5\right) - 3\right)\]

  6. Applied add-sqr-sqrt35.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\color{blue}{\left(\sqrt{w \cdot w} \cdot \sqrt{w \cdot w}\right)} \cdot \left(\sqrt{r} \cdot \sqrt{r}\right)\right) \cdot \left(\sqrt{r} \cdot \sqrt{r}\right), 4.5\right) - 3\right)\]

  7. Applied unswap-sqr35.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \color{blue}{\left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \left(\sqrt{w \cdot w} \cdot \sqrt{r}\right)\right)} \cdot \left(\sqrt{r} \cdot \sqrt{r}\right), 4.5\right) - 3\right)\]

  8. Applied unswap-sqr35.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \color{blue}{\left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \sqrt{r}\right) \cdot \left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \sqrt{r}\right)}, 4.5\right) - 3\right)\]

  9. Simplified35.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \color{blue}{\left(\left|w\right| \cdot r\right)} \cdot \left(\left(\sqrt{w \cdot w} \cdot \sqrt{r}\right) \cdot \sqrt{r}\right), 4.5\right) - 3\right)\]

  10. Simplified0.4

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left|w\right| \cdot r\right) \cdot \color{blue}{\left(\left|w\right| \cdot r\right)}, 4.5\right) - 3\right)\]
  11. Using strategy rm

  12. Applied fma-udef0.4

    \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + 4.5\right)} - 3\right)\]

  13. Applied associate--l+0.4

    \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)}\]
  14. Using strategy rm

  15. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)\]
  16. Using strategy rm

  17. Applied add-cbrt-cube7.1

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\color{blue}{\sqrt[3]{\left(\left(1 - v\right) \cdot \left(1 - v\right)\right) \cdot \left(1 - v\right)}}} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)\]

  18. Applied add-cbrt-cube21.4

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\frac{\color{blue}{\sqrt[3]{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right)\right) \cdot \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right)}}}{\sqrt[3]{\left(\left(1 - v\right) \cdot \left(1 - v\right)\right) \cdot \left(1 - v\right)}} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)\]

  19. Applied cbrt-undiv21.5

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\color{blue}{\sqrt[3]{\frac{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right)\right) \cdot \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right)}{\left(\left(1 - v\right) \cdot \left(1 - v\right)\right) \cdot \left(1 - v\right)}}} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)\]

  20. Simplified0.4

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\sqrt[3]{\color{blue}{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}}} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)\]

  21. Final simplification0.4

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\sqrt[3]{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}} \cdot \left(\left(\left|w\right| \cdot r\right) \cdot \left(\left|w\right| \cdot r\right)\right) + \left(4.5 - 3\right)\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))